Stat206: Random Graphs and Complex Networks Spring 2003 Lecture 2: Branching Processes Lecturer: David Aldous Scribe: Lara Dolecek Today we will review branching processes, including the results on extinction and survival probabilities expressed in terms of the mean and the generating function of a random variable whose distribution models the branching process. In the end we will brieﬂy state some more advanced results. Introduction Let’s start by considering a random variable X . If possible values of X are non negative integers, then for p i = P ( X = i ), the sequence ( p i ,i ≥ 0) denotes the distribution of X . Recall that E [ X ] = X i ≥0 ip i Some of the standard distributions and their parameters are: • Binomial ( n,p ), • Geometric ( p ), • Poisson ( λ ), • Normal ( μ,σ 2 ), and • Exponential ( λ ). Branching Processes Galton-Watson Branching Process In the Galton-Watson Branching Process (GWBP) model the parameter is a probability distribution ( p0 ,p 1 ,... ), taken on by a random variable ξ . For the GWBP model we have the following rule. Branching Rule : Each individual has a random number of children in the next generation. These random variables are independent copies of ξ and have a distribution ( p i ). Let us now review some standard results about branching processes. For simplicity let μ denote E[ ξ ]. Let Z n denote the number of individuals in the nth generation. By default, we set Z0 = 1, and also exclude the case when ξ is a constant. Notice that a branching process may either become extinct or survive forever. We are interested under what

This preview
has intentionally blurred sections.
Sign up to view the full version.